Let $M$ be a complex $n$-dimensional manifold and $R \subset M$ be a totally real, compact, $n$-dimensional (real) manifold. Let $\alpha$ be a smooth nonnegative $(n,n)-$form on $M$. Does there exist a smooth plurisubharmonic function $f \colon U \rightarrow \mathbb{R}$, where $U$ is a open neighbourhood of $R$ in $M$ such that the equation $(\partial \bar{\partial} f)^{n} = \alpha$ is satisfied? I am interested in a smooth solution, in a neighbourhood of the real manifold, of the inhomogenous Monge-Ampère equation. Can you give me a reference to say what can be done to have such a solution?
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$\begingroup$ I don't think there is much literature on this topic, unless you are willing to assume that $\alpha$ be nonnegative definite and seek for $f$ plurisubharmonic. $\endgroup$– YangMillsFeb 10, 2012 at 23:08
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$\begingroup$ yes ok then lets assume that ... i will edit the first post ! what about that ... can you offer me some literature ? $\endgroup$– williamFeb 11, 2012 at 6:12
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